I whitened by cartridge box sling today.
Mixed up some milk and chalk, as described in some historical manuals from around 1800. Looks pretty decent. To paint the sling I only used some ragged cloth, no brushes or anything else. The manuals do say that you should use a brush, but I can imagine that a lot of people would only have used some old cloth to paint it.
My idea is that I will whiten it now, and then let it get dirty for at least a year. I think that that will make the sling look much more "campaignish" than always whitening, or never whitening it.